1. Special Right Triangles

2. Draw 5 squares with each side length increasing by 1 1 2 3 4 5

3. Find the length of each diagonal 1 2 3 4 5

4. 1 a 2 + b 2 = c 2 1 2 + 1 2 = c 2 1 + 1 = c 2 2 = c 2  2 = c

5. Find the length of each diagonal 1 2 a 2 + b 2 = c 2 2 2 + 2 2 = c 2 4 + 4 = c 2 8 = c 2  8 = c 2  2 = c

6. Find the length of each diagonal 1 2 3 a 2 + b 2 = c 2 3 2 + 3 2 = c 2 9 + 9 = c 2 18 = c 2  18 = c 3  2 = c

7. Find the length of each diagonal 1 2 3 4 a 2 + b 2 = c 2 4 2 + 4 2 = c 2 16 + 16 = c 2 32 = c 2  32 = c 4  2 = c

8. Find the length of each diagonal 1 2 3 4 5 a 2 + b 2 = c 2 5 2 + 5 2 = c 2 25 + 25 = c 2 50 = c 2  50 = c 5  2 = c

9. Do you see the pattern 1 2 3 4 5 5  2 4  2 3  2 2  2  2 Side length time square root of 2

10. Now, what do you know about squares and the triangle formed by the diagonal. A square has 4 equal sides and 90° angles The triangle is isosceles right

11. If the triangle is isosceles right, then what do you know about the base angles? 45°

12. An isosceles right triangle is a 45-45-90 triangle 45°

13. 45-45-90 triangles follow the same pattern we found in the squares. 45° a a  2 So if you know one side you can find the other.

14. Find the missing sides 45° 8 8  2 c 8

15. Find the missing sides 45° a 7  2 7 7

16. Find the missing sides 25 25  2 c 25

17. Find the missing sides a 16  2 16

18. 45-45-90 triangles will have two basic diagrams that you will need to recognize. 45°

19. You will have to know 45-45-90 triangles… the hypotenuse = leg  2 45°

20. OR.. the leg = hypotenuse  2 45°

21. There is a second “special right triangle”…30-60-90 Obviously if one angle is 30° then the other is 60° 30° Angle Sum of Triangle

22. 30-60-90 triangle The short leg (a) is always opposite the 30° angle 30° a

23. 30-60-90 triangle The hypotenuse is always 2 times short leg 30° 2a a

24. Draw five 30-60-90 triangles Each short leg increasing by 1 30° 10 5 30° 8 4 6 3 4 2 2 1 Find the long leg b b b b b

25. a 2 + b 2 = c 2 1 2 + b 2 = 2 2 1 + b 2 = 4 b 2 = 4-1 b 2 = 3 b =  3 30° 2 1 b

26. Find the long leg a 2 + b 2 = c 2 2 2 + b 2 = 4 2 4 + b 2 = 16 b 2 = 16-4 b 2 = 12 b =  12 b = 2  3 30° 4 2 2 1 b b

27. Find the long leg a 2 + b 2 = c 2 3 2 + b 2 = 6 2 9 + b 2 = 36 b 2 = 36-9 b 2 = 27 b =  27 b = 3  3 30° 4 2 2 1 b b 6 3 b

28. Find the long leg a 2 + b 2 = c 2 4 2 + b 2 = 8 2 16 + b 2 = 64 b 2 = 64-16 b 2 = 48 b =  48 b = 4  3 30° 4 2 2 1 b b 6 3 b 8 4 b

29. Find the long leg 30° 4 2 2 1 b b 6 3 b 8 4 b 10 5 b a 2 + b 2 = c 2 5 2 + b 2 = 10 2 25 + b 2 = 100 b 2 = 100-25 b =  75 b = 5  3

30. Do you see the pattern? 30° 2 1 3 3  3 30° 4 4  3 30° 5 5  3 2  3  3 Long leg = short leg time the square root of 3

31. 30-60-90 triangle 30° 2a a a3a3

32. Find the missing side 30° h 8 8383 16 b